P-37 9 June 30 - July 3, 2015 Melbourne, Australia VORTICAL STRUCTURES AND PARTICLES IN RAYLEIGH-BENARD´ CONVECTION Sangro Park Department of Computational Science and Engineering Yonsei University 50 Yonsei-ro Seodaemun-gu [email protected] Changhoon Lee Department of Computational Science and Engineering & Department of Mechanical Engineering Yonsei University 50 Yonsei-ro Seodaemun-gu [email protected] ABSTRACT wall in experiments and explained life cycles of coherent Behavior of vortical structures and particle dispersion structures at the boundary layer where they strike the wall in Rayleigh-Benard´ convection is investigated by direct and excite waves. The waves propagate the line plumes numerical simulation using a spectral method. The flow which merge and release thermals to the opposite wall. The regime is soft turbulence with Rayleigh number of 106, physical properties of thermal plumes such as local heat Prandtl number 0.7 and the aspect ratio 6 : 1 : 6. Simul- flux and thermal dissipation rate were studied by (Shishk- tion reveals that the horizontal vorticity occurring near the ina & Wagner (2006);Shishkina & Wagner (2007);Shishk- wall and at the border of thermal plumes affect dynamics ina & Wagner (2008)). Inside the thermal plumes, local heat of thermal plumes significantly. Vortical natures of thermal flux, and vertical component of vorticity are large. In con- plumes are examined through the invariants of velocity gra- trast, heat flux value and thermal dissipation rate is small dient tensor. Inhomogeneous distribution of particles only in background region. At the border of the thermal plume, affected by fluid motions (one-way coupling) is also inves- thermal dissipation rate is large and horizontal components tigated using the point particle approach for Stokes number of vorticity become large. On the other hand, the vorti- 0.1, 1, 5 and 20. cal structures have been studied by the eigenvalues and in- variants of velocity gradient tensor and strain/rotation rate tensor for Isotropic and channel turbulent flows, but not in Rayleigh-Benard´ convection. (Ooi et al. (1999);Blackburn Introduction et al. (1996)) Rayleigh-Benard´ convection has a simple geometry so For the past few decades, the vast majority of studies of that the flow parameters can be easily controlled, while it Rayleigh-Benard´ convection have been addressing the sin- consists of various thermal flow dynamics. For this rea- gle phase cases. However, multiphase thermal convection son, many studies have focused on the global quantities deserves to be focused as a fundamental study of physics such as Nusselt number and its relation with flow param- and as a practical application in industry, weather predic- eters and geometry. Additionally, structures are known tion, etc. As a representative study of particle simulation to affect the Rayleigh and Prandtl number scaling so that study in Reyleigh-Benard´ convection, Oresta & Prosperetti many studies on visualization of the structures and physi- (2013) investigated thermal, mechanical and combined cou- cal properties of thermal structures have already been car- pling of fluid and particle motions in slender cylinder. They ried out. The predominant structures near the wall, in revealed the effect of particles on heat flux and turbu- which large thermal gradient and conduction heat flux ex- lence intensity in thermal convection. Particle studies in ist, are thermal and line plumes. Thermals are hot/cold Rayleigh-Benard´ convection in a long cylinder with relation fluid blobs released and detached from the boundary layer. to thermal structures are carried out in Oresta et al. (2006). Krishnamurti & Howard (1981) modeled the near-wall dy- They investigated the invariants of velocity gradient tensor namics as a periodic growth of conduction layer by diffu- at particle locations, and showed the features in particle ac- sion, which becomes unstable and releases thermals when cumulation regions. Although studies of particle behavior Rayleigh number based on the conduction layer thickness in turbulence are vigorously conducted in various geome- becomes about 1000; then new conduction layer begins tries like pipe, channel and Isotropic domains (Rouson & to form and grow. Line plumes are structures which re- Eaton (2001);Elghobashi (1994)), particle studies in natural lease hot/cold fluid from lines continuously. They move convection in a wide plane domain is rare. randomly in space and merge with one another. Zocchi et al. (1990) visualized propagating line plumes near the In this study, we mainly investigate the relation be- 1 tween vortical structures and thermal plumes by focusing Invariant analysis of vortical structures on vertical stretching and horizontal shear near the wall. Flow domain is divided into three regions (bound- Additionally, we investigated the topology of vortical struc- ary layer, plume mixing layer and central region), which tures with joint probability density functions of invariants show distinguishing behaviors. Inside the thermal bound- of velocity gradient tensor. For the particle simulation we ary layer, conductive heat transport is dominant mecha- considered the effect of fluid to particle motion (one-way nism. Due to the horizontally moving sheet-like plumes, coupling) using the point particle approach. The simulation horizontal shear is large, which makes horizontal stretch- results include time averaged concentration of particles of ing of fluid element. Central region covers mid-plane and various Stokes number in the vertical locations in the do- shows nearly isothermal behaviors. At the thermal plume main. mixing layer, flows from boundary layer matches to the cen- tral region. In this paper, the three regions are classified based on the vertical distribution of mean temperature and vertical acceleration. Inside boundary layer region is lo- Numerical Approach cated at 0 < y < δT /H(= 0.0375), plume mixing layer is Rayleigh-Benard´ convection is defined as flow con- 0.04 < y < 0.3, and the central region is at 0.3 < y < 0.5 for fined between hot lower boundary wall and cold upper one. the lower half region of the whole flow domain. In this study, the dimensions of horizontal (x , z ) and − − Velocity gradient tensor, Ai j = ∂ui/∂x j has the charac- vertical(y ) directions are L : L : L = 6 : 1 : 6. Con- − x y z teristic equation given by stant temperatures and no-slip velocity conditions are used at boundary walls and periodic boundary conditions for side walls. A spectral method with dealiased Fourier and 3 2 λ + PAλ + QAλi + RA = 0 (5) Chebyshev expansions in the horizontal and vertical direc- i i tions are used to solve the governing equation. The Crank- Nicolson method and a 3rd-order Runge-Kutta schemes are Here λ is eigenvalue and the invariants can be derived as applied for time advancing of the viscous terms and nonlin- i following equations. ear terms including buoyant source, respectively. The gov- erning equations are non-dimensionalized by using κ/H, H, ∆T(= 2Tw), where κ, H, ∆T, Tw are thermal conduc- P = A (6) tivity, domain height, temperature difference, wall temper- A ii ature, respectively. The non-dimensional governing equa- 1 QA = Ai jA ji (7) tions are − 2 1 R = A A A (8) A − 3 i j jk ki ∂u i = 27 2 3 0 (1) DA = RA + QA (9) ∂xi 4 2 ∂ui ∂ui ∂ p ∂ ui + u = + Pr + RaPrT 0δ (2) ∂t j ∂x − ∂x ∂x ∂x i2 j i j j The joint probability density functions of the invariants 2 ∂T ∂T ∂ T shows the topology of the local vortical structures. Figure + u j = (3) ∂t ∂x j ∂x j∂x j 1 illustrates the joint probability density functions between RA and QA inside thermal boundary layer and central re- gions. SF/S, UF/C indicate stable focus/stretching, unstable In equation (2), Rayleigh and Prandtl numbers are defined focus/contraction at positive determinant region (DA > 0). as Ra = (βgH3∆T)/(κν), Pr = ν/κ and β, ν are thermal SN/S/S, UN/S/S mean stable node/saddle/saddle, unstable expansion coefficient and kinematic viscosity. node/saddle/saddle at negative determinant region. Like A one-way particle tracking algorithm is coupled with other turbulent flows, zero at the origin is the most proba- fluid governing equations to calculate particle trajectories ble value and shows preference along DA = 0 line in fourth in the flow field using the 4th-order Hermite interpolation quadrant, which indicates most of the largely strained ed- combined with 6th-order Lagrangian polynomial. With dies show near-vortical motion, i.e., predominance of vor- the known fluid velocity at particle position, the equations tex stretching. Near the wall, the probability in region of of particle motions are integrated employing a 3rd-order DA < 0 increases. These regions are convergence regions, Runge-Kutta scheme. The dimensional governing equation i.e., along two principal axes flow converges and flow di- of particles (ignoring gravity effect) are verges in the other axis. This seems to be attributed to wave propagation from thermal plumes. The shapes of joint prob- ability density functions at different regions of Rayleigh- dv f (Rep) Benard´ convection are mostly affected by thermal plume = (u v) (4) dt τp − activities. As explained in Zocchi et al. (1990), vertically moving thermal plumes make spreading and concentration of fluid at boundary layer regions by striking the wall, there- In the equation, the particle Reynolds number is defined as fore horizontally propagating sheet-like thermal structures Re = (d v u )/ν, where d , v, u are particle diame- are dominant at boundary regions as shown by preference to p p| − | p ter, particle velocity, and fluid velocity at particle location. UN/S/S region in figure 1 (a). Although vertical movements 2 τp is the particle response time defined as (ρpdp)/(18ρν). of thermal plumes, nearly isothermal behavior is shown in Stokes numbers Stk, the ratio of particle response time to the central region because of mixing.